A signal is a physical phenomenon distributed over space and/or time. Examples include signals distributed over time, such as electromagnetic waves on antennas or transmission lines; signals distributed over Fourier space, such as optical or electrical spectra; and multidimensional signals distributed over physical space, such as 2 D and 3 D images.
In digital signal analysis, a signal is reconstructed from discrete measurements. For many years, sampling theory formed the theoretical core of signal analysis. Conventional approaches to sampling signals or images follow Shannon's celebrated theorem that the sampling rate must be at least twice the maximum frequency present in the signal. This minimal sampling rate is termed the Nyquist rate or frequency. In fact, this principle underlies nearly all signal acquisition protocols used in consumer audio and visual electronics, medical imaging devices, radio receivers, etc. In the field of data conversion for example, standard analog-to-digital converter (ADC) technology implements the quantized Shannon representation that the signal is uniformly sampled at or above the Nyquist rate.
In many applications, including digital image and video cameras, the Nyquist rate is so high that too many samples result, making compression a necessity prior to storage or transmission. In other applications, including imaging systems (medical scanners and radars) and high-speed analog-to-digital converters, increasing the sampling rate is very expensive.
Recent developments have shown that compressive sampling or compressive sensing can provide sub-Nyquist rate sampling. However, a number of challenges exist in implementing such sampling algorithms and reconstructing the original signal without a significant loss of data. These and other limitations are overcome in embodiments of the invention described herein.